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I’m trying to learn about calculating coordinates for $\theta$ and $\varphi$ in a Bloch-sphere.

I came accross this book about it, including example questions.

At question 2.12b, they ask to give the value for theta and psi for the state |i⟩. In the answers they rewrite the formula down below.

Can anyone explain where the $\frac{1}{\sqrt{2}}$ comes from?

enter image description here

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    $\begingroup$ The book has a section about all states being of unit length. I suggest to read it one more time but carefully. $1/\sqrt2$ makes sure the state is on the Bloch sphere and not outside or inside of it. $\endgroup$
    – MonteNero
    Dec 30, 2022 at 20:17

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The coefficients come from normalization condition. It ensures that sum of probabilities of measuring basis states forming a quantum state is equal to one. The coefficients are so-called probability amplitudes, not probabilities themselves. It holds that probability of measuring $i$th basis state is $|a_i|^2$, where $a_i$ is probability amplitude of the basis state. As mentioned above, it must hold that $\sum_i |a_i|^2 = 1$.

In your case, you have two basis states, namely $|0\rangle$ and $|1\rangle$. Probability of measuring each of them is 50% as $1/\sqrt{2}$ squared is $1/2$.

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